GEOPHYSICAL RESEARCH LETTERS, VOL. 30, NO. 24, 2255, doi:10.1029/2003GL018498, 2003
Spontaneous generation and impact of inertia-gravity waves in a
stratified, two-layer shear flow
P. D. Williams and P. L. Read
Atmospheric, Oceanic and Planetary Physics, Clarendon Laboratory, University of Oxford, UK
T. W. N. Haine
Department of Earth and Planetary Sciences, Johns Hopkins University, Baltimore, USA
Received 26 August 2003; revised 12 November 2003; accepted 17 November 2003; published 19 December 2003.
[1] Inertia-gravity waves exist ubiquitously throughout
the stratified parts of the atmosphere and ocean. They are
generated by local velocity shears, interactions with
topography, and as geostrophic (or spontaneous)
adjustment radiation. Relatively little is known about the
details of their interaction with the large-scale flow,
however. We report on a joint model/laboratory study of a
flow in which inertia-gravity waves are generated as
spontaneous adjustment radiation by an evolving largescale mode. We show that their subsequent impact upon the
large-scale dynamics is generally small. However, near a
potential transition from one large-scale mode to another, in
a flow which is simultaneously baroclinically-unstable to
more than one mode, the inertia-gravity waves may strongly
influence the selection of the mode which actually
INDEX TERMS: 3210 Mathematical Geophysics:
occurs.
Modeling; 3220 Mathematical Geophysics: Nonlinear dynamics;
3346 Meteorology and Atmospheric Dynamics: Planetary
meteorology (5445, 5739). Citation: Williams, P. D., P. L.
Read, and T. W. N. Haine, Spontaneous generation and impact of
inertia-gravity waves in a stratified, two-layer shear flow,
Geophys. Res. Lett., 30(24), 2255, doi:10.1029/2003GL018498,
2003.
1. Introduction
[2] Like many physical systems, fluids often exhibit the
coexistence of motions on a wide range of space and time
scales. Correspondingly, the linear normal modes of the
governing Navier-Stokes equations generally have spatiotemporal structures which fall naturally into distinct classes,
when categorized according to the fundamental dynamical
mechanisms which permit their existence. This property of
the fluid equations was first identified by Margules [1893],
who derived two species of solutions to Laplace’s tidal
equations. He named his solutions ‘‘Wellen erster Art’’
(waves of the first type) and ‘‘Wellen zweiter Art’’ (waves
of the second type), which we now know more familiarly as
inertia-gravity and Rossby waves.
[3] Characteristic wavelengths, intrinsic frequencies and
propagation speeds of inertia-gravity and Rossby waves
differ by at least an order of magnitude, in both the
atmosphere and ocean. A convenient, often tacit, assumption is that mutual nonlinear interactions between the scaleseparated modes are negligible. This justifies the use of a
Copyright 2003 by the American Geophysical Union.
0094-8276/03/2003GL018498$05.00
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reduced-dimensional description of the fluid system based
upon a projection onto its slow manifold [Leith, 1980], a
sub-surface of the full phase space upon which inertiagravity mode amplitudes are identically zero. Benney [1977]
has shown that a resonant triad interaction between two
short waves and one long wave is possible if the phase
speed of the long wave is equal to the group speed of the
short waves, however. The physical interpretation is that the
energy of the short modes, which travels at their group
speed, does not drift relative to the phase of the long mode,
a requirement which evidently allows a resonant reinforcement of the energy transfer. The typical dispersion curves
plotted in Figure 1 show that—in principle, at least—the
phase speed of a Rossby wave can match the group speed of
a gravity wave. This challenges the assumption that the
wave-wave interaction is always negligible.
[4] Laboratory observations of systematic inertia-gravity
wave generation by evolving quasi-geostropic modes were
reported by Lovegrove et al. [2000]. This finding meant that
the interaction could be investigated in a real fluid, without
recourse to the ad hoc theoretical idealizations typically
made in studies of highly truncated models [e.g., Lorenz,
1986]. The present study is an extension of the work of
Lovegrove et al. [2000] (Section 2). We employ a highresolution numerical model to investigate the mechanism
by which the inertia-gravity waves are generated in the
laboratory experiment (Section 3), and then we incorporate
a stochastic parameterization of the inertia-gravity waves
into the model in order to assess their impacts upon the
large-scale flow (Section 4). We end with a statement of our
conclusions and a brief discussion in Section 5.
2. The Rotating, Two-Layer Annulus Experiment
[5] In the present laboratory experiments, the fluid occupies the annular domain defined in cylindrical coordinates
(r, q, z) by 0 < z < 25.00 cm, 6.25 cm < r < 12.50 cm and
0 < q 2p. The domain is filled with equal volumes of two
immiscible liquids (water, and a mixture of d-limonene and
CFC-113) whose densities differ by around 0.5%, to give a
well-defined equilibrium interface at z = 12.50 cm. The base
at z = 0 and sidewalls at r = 6.25 cm, 12.50 cm are made to
rotate about the axis of symmetry under computer control,
and the lid at z = 25.00 cm is made to rotate relative to the
base. This differential lid rotation provides the velocity
shear across the interface required for the generation of
a large-scale mode due to baroclinic instability, if the
background rotation is large enough. For the experiments
3
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WILLIAMS ET AL.: GENERATION AND IMPACT OF GRAVITY WAVES
Figure 1. Schematic dispersion curves for three different
zonally-propagating shallow-water wave modes, showing
intrinsic angular frequency w as a function of zonal (EastWest) wavevector kx. The Rossby wave chord slope at the
long wavevector kl is equal to the inertia-gravity wave
tangent slope at the short wavevector ks, leading to the
possibility of a resonant triad interaction (see text). Acoustic
modes are too fast to take part in such an interaction, which
justifies filtering acoustic modes from general circulation
models.
described herein, the typical Froude number, Rossby
number, Reynolds number, Ekman number and interfacial
tension number are F 10, Ro 0.1, Re 103, Ek 106
and I 0.1, respectively [Williams, 2003].
[6] The apparatus is viewed from above by a colour
video camera on the axis of symmetry around 2 m above
the annulus, and is illuminated from below by a bright white
lamp. The base, lid and fluids are transparent and colourless, allowing the passage of light through the apparatus.
The system is viewed through crossed polaroids, and
the lower-layer liquid is optically active, giving a direct
relationship between colour observed by the camera and
lower layer depth (see Hart and Kittleman [1986] for more
details).
[7] A typical still from the video footage is shown in
Figure 2. The large-scale wave, of azimuthal wavenumber 2,
has arisen from a baroclinic instability and drifts slowly
around the annulus. Two trains of deep-water inertia-gravity
waves (IGWs) have developed and are superimposed in
the troughs of the shallow-water large-scale mode. Their
wavelengths, intrinsic periods and amplitudes are each a
factor of around 10 smaller than those of the large-scale
mode. The structure of the IGW trains bears a striking
resemblance to those generated during the spontaneous
adjustment process in the high-resolution atmospheric
simulations of O’Sullivan and Dunkerton [1995], despite
the differences in length and time scales between the
atmosphere and laboratory.
3. Numerical Annulus Model
[8] There are two candidate generation mechanisms for
the IGWs observed in the laboratory, as in the free atmosphere. The first is the spontaneous emission radiation
mechanism [Ford et al., 2000], and the second is a
Kelvin-Helmholtz shear instability. In order to examine
which of the two mechanisms may be responsible in the
laboratory, the horizontal velocity fields are needed, which
are unavailable from the experiment. To this end, we
simulate the large-scale laboratory flow using a two-layer
quasi-geostrophic annulus model known as QUAGMIRE
[Williams, 2003]. The model integrates the quasi-geostrophic potential vorticity equations in both layers, with
parameterized Ekman pumping and suction velocities at
the lid, base and interface. Cylindrical geometry is used,
and the effects of interfacial tension are included. We use a
grid of 96 points in azimuth and 16 in radius. We perform
the timestepping using the potential vorticity tendency
equations in physical space, but transform to normal mode
space once per timestep to obtain the streamfunction
by inverting the potential vorticity. At the sidewall
boundaries, we impose impermeability on the eddy flow
components, and no-slip boundary conditions on the
mean-flow correction component.
[9] Ageostrophic IGWs are filtered out of the model by
construction, and so the model velocity fields allow us to
assess the incipient generation of IGWs by a pure quasigeostrophic mode. A typical model lower layer depth field
is shown in Figure 3a, for comparison with the laboratory
image in Figure 2 which was obtained with similar background and differential rotation rates. The model wave
amplitudes, phase speeds, and zonal wavenumbers agree
reasonably well with those observed in the laboratory
[Williams, 2003].
[10] Ford [1994] derived an equation which indicates the
local strength of spontaneous emission radiation in barotropic shallow water:
@2
@
g@ 2 2
2
2 @h
þ
f
gHr
¼ r :F þ f k:r
rFþ
r h ;
@t 2
@t @t
2 @t
ð1Þ
where
F ¼ ur
r:ðhuÞ þ ðhu:r
rÞu:
ð2Þ
Figure 2. Typical image captured by a video camera
viewing the laboratory annulus from above. In the original
colour image, the high intensity region near the inner
sidewall appears blue and corresponds to an elevated lower
layer depth of around z = 13 cm, and the high intensity
region near the outer sidewall appears yellow and
corresponds to a reduced depth of around z = 11 cm. The
relatively dim region in between appears red and corresponds to lower layer depths close to the undisturbed value
of z = 12.5 cm.
WILLIAMS ET AL.: GENERATION AND IMPACT OF GRAVITY WAVES
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in magnitude should also be regions of significant IGW
production by the spontaneous emission mechanism. By
comparing Figures 2 and 3b we see that the magnitude of
LRT has maxima at just those regions where we observe
laboratory IGWs. There are smaller, subsidiary maxima in
LRT at other locations, but presumably LRT has to be large
enough to overcome dissipative effects—which are not
included in Ford’s theory—before generation actually
occurs. The correlation between regions of laboratory
IGW production and regions of subcritical model Richardson number is poor, as shown in Figure 3c. The Richardson
number drops below unity only near the outer sidewall,
which is not an observed generation region. We conclude,
therefore, that spontaneous emission, rather than a KelvinHelmholtz shear instability, is responsible for the production
of the observed laboratory IGWs. This conclusion has been
verified at a range of Froude numbers (from 5 to 20),
corresponding to a spectrum of different dominant azimuthal wavenumbers (1, 2 and 3).
[12] In the classic Rossby geostrophic adjustment problem, two IGW trains are emitted in opposite directions so
that the total ageostrophic component is zero at the source
region. In our experiment, however, all waves are superimposed on a strong zonal flow. Any IGWs which are
emitted in the retrograde direction (i.e., against the background flow) can therefore appear to propagate in the
prograde direction when viewed by our camera in the frame
of the annulus base. This is consistent with our observations,
which show a single train of unidirectional IGWs (e.g.,
Figure 2). The Rossby-Kelvin instability identified by Sakai
[1989] can be discounted as the cause of the IGW emission
we observe, since this resonant instability occurs only for
Froude numbers F 0.7, which is much smaller than the
Froude numbers achieved in the current study.
Figure 3. (a) Lower layer depth, (b) Lighthill Radiation
Term magnitude, and (c) Richardson number, from a
simulation of an azimuthal wavenumber 2 flow corresponding to the laboratory image of Figure 2. The quasigeostrophic annulus model described in Section 3 was used
to produce these diagrams.
Here, f = 2
is the Coriolis parameter, g is the acceleration
due to gravity, h is the instantaneous layer depth, with a
mean value of H, u is the horizontal velocity field, and k is
the unit vertical vector. The left side of equation (1) is the
linear IGW operator acting on @h/@t. The right side contains
all of the nonlinear terms, which we refer to collectively as
the Lighthill Radiation Term (LRT), as Ford’s analysis was
an extension of the work of Lighthill [1952] on the
generation of acoustic modes. Though derived using a
shallow water assumption, we assert that the same LRT
formula applies to the deep water limit, which is more
appropriate for the short waves in the laboratory. This is
reasonable, since whether or not an evolving, large-scale
mode will generate IGWs is not expected to depend upon
whether they would propagate in deep or shallow water, if
emitted.
[11] Figure 3b shows a plot of LRT for the lower annulus
layer, derived from the model velocity fields and lower
layer depth at the time of the flow in Figure 3a (g is replaced
with the reduced gravity g0 in (1) to account for the
stratification). Ford argued that regions where LRT is large
4. Stochastic Resonance
[13] To explore the impact of the IGWs on the large-scale
flow we perform a second model calculation, this time with
a random small-amplitude anomaly added to the right side
of the model quasi-geostrophic potential vorticity equations
for both layers. These new terms are intended to mimic the
potential vorticity anomalies induced by the IGWs in the
laboratory, which occur almost at the scale of the model
gridspacing. Unlike in the laboratory, the stochasticallyparameterized IGWs are global, i.e., they are included
everywhere, not just in those places at which LRT is large.
This may strengthen the correspondence between the
model, and the atmosphere and ocean, in which IGWs are
perhaps more ubiquitous than in the laboratory. At each
gridpoint and timestep, a random number is drawn from a
uniform distribution, with a width chosen such that the
resulting small-scale interface perturbations have a rootmean-square amplitude similar to those in the laboratory
(i.e., around one millimetre). The effects of the parameterized IGWs on the evolution of the large-scale mode are
generally small: wave speeds and wavenumbers are unaltered, and there is little change in amplitude.
[14] In some cases, however, the IGWs were found to
exert a strong influence on the balanced flow. To illustrate
this, a model run was performed with the amplitude of the
noise terms slowly increasing with time, after allowing an
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WILLIAMS ET AL.: GENERATION AND IMPACT OF GRAVITY WAVES
Figure 4. Hovmüller diagram, showing a mid-radius
azimuth-time contour plot of perturbation potential vorticity
in the lower layer, at the time of a spontaneous transition
from azimuthal wavenumber 2 to 1.
azimuthal wavenumber 2 mode to equilibrate in the absence
of noise. When a critical noise amplitude was reached—
which was still small compared with the amplitude of the
background large-scale mode—a spontaneous transition
occurred to a large-scale mode with azimuthal wavenumber 1.
Potential vorticity as a function of time and azimuthal angle,
at the time of the transition, is shown in Figure 4. Such
transitions were never observed in the absence of the
stochastic forcing, and were not reversed if the noise was
gradually decreased back to zero.
[15] These spontaneous transitions occurred when the
linear growth rates of the two large-scale modes involved
in the transition were similar, i.e., the system was close to a
parameter space wavenumber transition curve in a region of
intransitive multistability. The phenomenon which allows a
small (stochastic) forcing to produce a large (resonant)
response is known as stochastic resonance, and has been
observed before in numerical models of rapidly-rotating
barotropic fluid systems [e.g., De Swart and Grasman,
1987].
[16] Further evidence to support these model findings
comes from the present laboratory experiments. By varying
the interfacial tension between the two layers, using a
surfactant, we have been able to run experiments in which
the large-scale baroclinic mode is not significantly altered,
but the small-scale IGWs are greatly suppressed. This is
possible because the effects of interfacial tension are scaleselective, affecting the dynamics of short scales much more
than long scales. In such laboratory experiments, the largescale flow was found to exhibit a reluctance to undergo
wavenumber transitions when the flow was devoid of
IGWs, but more readily underwent such transitions when
IGWs were present.
5. Conclusions
[17] We have observed the spontaneous generation of
inertia-gravity waves by an evolving large-scale flow in the
laboratory. The times and spatial locations at which inertiagravity waves were observed appear to be well-predicted by
the formula due to Lighthill and Ford, and very poorly
predicted by analyses based on a critical Richardson number. Shear instability therefore seems unlikely as the source,
and we conclude that the observed short-scale waves are
most likely generated as spontaneous emission radiation by
the large-scale mode.
[18] We have investigated the impact of the short waves
on the long waves by using a numerical model of the large-
scale flow with a stochastic inertia-gravity wave parameterization. In general, the impact is small, but we have
identified circumstances in which stochastic resonance
allows small-amplitude inertia-gravity waves to interact
nonlinearly with the balanced flow in a profound way, by
forcing spontaneous transitions between balanced modes
and influencing long-term mode selection.
[19] The large-scale waves in the laboratory are prototypes of synoptic tropospheric and mesoscale oceanic
disturbances. It is therefore likely that small-amplitude,
small-scale inertia-gravity waves in the atmosphere and
ocean might resonate stochastically to force significant
changes to the large-scale balanced flow. This finding adds
to the evidence [e.g., Palmer, 2001] that deterministic
parameterizations of inertia-gravity waves in weather and
climate models may be unable to capture the full details of
the nonlinear interactions. The present results suggest that
certain aspects of the interaction can only be captured by a
stochastic parameterization or, presumably, by an explicit
representation of the inertia-gravity waves. There is,
therefore, a strong case for carrying out further research
regarding stochastic representations of sub-grid scale
processes in general circulation models.
[ 20 ] Acknowledgments. PDW acknowledges support under a
research studentship from the UK Natural Environment Research Council,
held at the University of Oxford, with award reference number GT04/1999/
AS/0203.
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P. D. Williams and P. L. Read, Atmospheric, Oceanic and Planetary
Physics, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, UK.
([email protected])
T. W. N. Haine, Department of Earth and Planetary Sciences, Johns
Hopkins University, Baltimore, USA.